You can still follow parts of route 66 after Flagstaff, there is Williams, Seligman and King man that can all be seen after the Grand canyon.

So if I am correct you have 31 days for the road trip out and back from Chicago including the National parks ? [not including 5 days in SF] Here is a rough sketch of what your trip could then look like.

Out.

Day 1. Chicago to Springfield IL.
Day 2 Springfield to St Louis.
Day 3. St Louis to Arcadia. [[Tulsa ?]

Day 4. to Oklahoma.
Day 5. Oklahoma to Amarillo.
Day 6 Amarillo to Albuquerque.
Day 7 Albu to Winslow.
Day 8. Winslow to Monument Valley.
Day 9 and 10. Grand canyon. [to Flagstaff]
Day11 Flagstaff to Williams/Seligman and Kingman.
Day 12 Kingman to Las Vegas via Hoover Dam.
Day 13 Las Vegas.
Day 14 Las Vegas to Bishop via Death valley.
Day 15 and 16 Yosemite.
Day 17 San Fran [5 days]

Back.

Day 18 SF to Lake Tahoe
Day 19 Lake Tahoe to Ely or Baker. [Great basin NP]
Day 20 Great Basin to Zion NP.
Day 21 Zion.
Day 22 Zion to Bryce canyon
Day 23 Bryce canyon to Capital Reef NP. [UT scenic 12]
Day 24 CR [Torrey] to Moab UT
Day 25 Arches and Canyonlands. [Moab UT]
Day 26 Moab to Durango CO [Mesa Verde NP?]
Day 27 Durango to Montrose. Million dollar Highway. [Silverton and Ouray on way]
Day 28 Montrose to Canon City. [Black canyon, Currecanti Nat res, Monarch pass, Royal Gorge bridge]
Day 29 Canon City to Dodge City.
Day 30 Dodge City to Kansas City.
Day 31 back to Chicago

This gives you all your time in the City's and some breath taking National parks and scenic drives. Of course this is just a quick and rough guide to what is possible and will hopefully give you something to work with and then adjust as needed to suit your own personal goals.